I'm looking for the name of what I suspect must be a standard property, and also for a possible statement about that property. 

First the property: $W=a_0\ldots a_{n-1}$ has *this property* if for all $1\le k<n$, 
$a_0\ldots a_{k-1}\ne a_{n-k}\ldots a_{n-1}$. 

In particular, this implies that in any finite or infinite word, the blocks containing $W$ are disjoint. For the particular application that I have in mind, I start off with an infinite word $x$, and replace some subwords of $X$ of length $n$ spaced far apart by $W$'s. The consequence of the definition that is useful for me is that if $x$ initially contained no $W$'s, then the only $W$'s in the resulting sequence are those $W$'s that I "manually" inserted. 

<blockquote>
Is there a name for this property?
</blockquote>

<blockquote>
Is it the case that if $X$ is any mixing shift of finite type, then $X$ contains a word with this property?
</blockquote>
(It's not hard to show that if $X$ is a full shift with an alphabet with two or more symbols, then $X$ contains words with this property.)